CIRCLES C2

In the following figure, prove that the angles in the same segment of a circle are equal.

Solution

Given : A circle with centre O and the angles ∠USV and 
               ∠UTV in the same segment formed by 
                the chord UV (or arc UAV)

Required to prove: ∠USV = ∠UTV

Construction : Join OU and OV.

Proof :

Try to remember that angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle, hence we have,

∠UOV = 2 ∠USV ...............(i)

and,

∠UOV = 2∠UTV …………...(ii)


Since (i) = (ii), we get,

2∠USV = 2∠UTV

2∠USV = 2∠UTV                 (dividing by 2 both sides)
2              2

∠USV = ∠UTV

∴ ∠USV =∠UTV   proved!